Because fractions tend to cause trouble for many people let us delve into them
Let us look at the fraction one fourth. We don’t write it one/fourth but rather ¼. In calculating instead of using the words one, two, three, four, five, sic, etc. we use these marks, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The Romans and Greeks borrowed symbols for calculating their alphabet. Our symbols come from the alphabet of an Oriental people. They are used because it is quicker and more convenient to use them than to multiply say, one hundred thirty-three by nine. It can be done though.
Nine times one hundred is nine hundred.
Nine times thirty is two hundred seventy.
Nine times three is twenty-seven.
The sum is eleven hundred ninety-seven.
Therefore it is quite easy to see why we have borrowed these symbols which come to us through the Arabs.
A possible reason for difficulty with fractions is that the form does not make for simpler computation as we have seen in multiplication. The person who is not sure what must be done in adding fractions should make a problem using words instead of Arabic numbers. For example: add one fourth plus three fourths. This answer is gotten the same way as if they added one pencil plus three pencils. A person after answering either one of these problems can easily see how to do ¼ ¾.
You can do multiplication the same way. The problem ½ x ¾ is written one half times three fourths. The one and three are multiplied together because they both are whole numbers. The product of that is three. Then the fractions, the half and the fourth, are taken together. If you want to say a half times a fourth gives an eighth I would have no objections. Actually you should say a half of a fourth is an eighth. From this you should see how to calculate ½ x ¾.
In division you are taught the rule that you must invert the second term and multiply. And of course after that the same methods could be used as was seen before for multiplication.
Division of fractions can be done not using the rule of inverting and multiplying. Just that it is easier to make young students memorize the rule than to make them understand the methods of doing division.
It is very fascinating to do division problems by their actual method rather than following a rule but some other concepts must be brought in which would make this paper lengthy.
Therefore in conclusion I hope you enjoyed reviewing these fraction concepts as much as I enjoyed writing about them.

Originally Published in:
Vol. 19 No. 5 June-July 1959